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Overview on a selection of recent works in asymptotic analysis for wave propagation problems
Xavier Claeys
In recent years, the study of wave propagation has induced a lot of research in relation with asymptotic analysis. While many types of problems may possibly be linked both to asymptotic analysis and scattering theory, the present overview focuses on recent advances concerned with wave propagation problems involving a small perturbation of geometrical nature.
General issuesConsider a well posed wave propagation problem (Pδ) with
solu-tion uδ (called ”exact”) that depends on a parameter δ satisfying kδ << 1 where
k is the wave number. The problem (Pδ) is assumed to be a perturbed version
of a limit problem (P0) from which it differs only in a localised region that is as
small as δ in one (or more) direction of space.
Our purpose is to design a numerical method for (Pδ) that would keep track of
the influence of the perturbation at a reasonnable computational price. One idea is to find another well posed problem (ePδ) that satisfies two features:
• its dependency with respect to δ is easier to handle numerically, • its unique solution euδ is a sufficiently sharp approximation of uδ.
Deriving such an approximate model for (Pδ) is usually not a trivial issue. Suppose
that we want to build (ePδ) such that kuδ−euδk = O( η(δ) ) with η(δ) → 0. Consider
the expansion
uδ(x) = u0(x) + λ1(δ)u1(x) + · · · + λN(δ)uN(x) + O( η(δ) )
On possible approach consists in identifying a (well posed) problem (ePδ) that would
be satisfied byPNn=0λn(δ)un(x) up to a residual in O(η(δ)), and proving that euδ
satisfies a relevant error estimate noting that euδ=PNn=0λn(δ)un+ O( η(δ) ).
Although approximate models can be easier to handle, they may contain ex-otic features which would require specific numerical treatments. Once a numerical method has been proposed for ePδ, a typical issue consists in proving that there
exists C > 0 (independent of δ) and µ(h) → 0 such that keuδ− euδ,hk ≤ C µ(h) for
all h, δ ∈ (0, 1) where euδ,h is the discrete solution.
Thin layer problems In the first category of problem that we examine, the perturbation has the structure of a thin layer of thickness δ located along smooth boundaries or smooth interfaces of the domain of propagation. In such a case the exact solution admits an expansion of the form: uδ = u0+ δu1+ δ2u2+ . . . .
A possible approximate model then consists in posing the wave equation in the limit geometry, as if there were no perturbation, and taking into account the thin layer by means of modified boundary conditions called Generalized Impedance Boundary Conditions (GIBC).
Scattering by objects with thin dielectric coating is a first exemple of such a case. On this subject, Engquist and N´ed´elec in [15] proposed a GIBC that was much sharper than any of the modified impedance boundary conditions that had been previously introduced. Since then, a complete theoretical framework for thin coatings has been developped by Bendali, Lemrabet, Joly, Haddar and co-workers in [2, 3, 4, 14, 16], including derivation of full expansions and rigorous justifications of GIBCs of high order. Besides Poignard in [29] studied the case a high contrasted thin coating, and Chun and Hestaven in [7] validated the good computational efficiency of GIBCs in a discontinuous Galerkin context.
Scattering by a highly absorbing obstacle is another case of thin layer problem. Assume that such an object admits an absorption coefficient σ = k/δ2
where δ → 0. A wave penetrating such an object keeps a significant amplitude only in a region of thickness δ concentrated along the exterior boundary of the obstacle: this is the skin effect. Whereas thin coating problems had been studied for long, there existed very few works on high absorption problems until recently, except [1, 32]. Haddar, Joly and Nguyen in [17, 18] proposed results comparable to what had been established in the case of thin coatings. P´eron, Dauge and co-workers in [6, 27] also proposed a derivation of full expansions for obstacles with Lipschitz boundary (not just smooth) and studied precisely the influence of the geometry on the skin thickness. Finally Haddar and Lechleiter in [19] proposed a similar analysis in the context of scattering by an unbounded obstacle.
The case of a domain of propagation containing a dielectric layer of thickness δ is a third example that caught attention only recently. In [31] Schmidt and Tordeux proposed a full expansion and approximate trasmission conditions pre-cise in O(δ2
) for a scalar problem. Poignard and P´eron in [28, 30] derived an expansion in O(δ3
) of the exact solution to an electromagnetic scattering problem. Chun, Haddar and Hestaven in [8] formally derived an expansion for a time do-main electromagnetic problem and studied computational efficiency of high order GIBCs in a discontinuous Galerkin framework.
Geometric singular perturbation problemsIn a second part of this presenta-tion we focus on asymptotic problems for which the solupresenta-tions admit an expansion with terms that may have a singular behaviour related to a singularity appearing in the geometry. In this situation, the exact solution may admit an expansion with more complex structure than for thin layer problems. On this type of as-ymptotic analysis for elliptic problems, the reference book is [26]. For such cases different asymptotic approaches are possible, we comment on two: multiscale ex-pansion method and matched asymptotics. Clear presentation and comparison of those two approaches was the subject of [13]. Concerning matched asymptotics, a reference book is [20].
Multiscale expansion method and matched asymptotics share several common features. Both techniques involve far field functions expressed in standard vari-ables, and near field functions that depend on scaled varivari-ables, and in both cases the expansion is obtained as an interpolation between the near and the far field
terms. These methods differ on the cutt-off functions used to interpolate, and on the exact procedure used to construct the far field and near field terms. The mul-tiscale expansion method provides sharper approximations but looks less intrinsic than matched asymptotics. The construction procedure associated to matched asymptotic requires to enforce an algebraic identity called ”matching principle”.
Several recent works dedicated to problems of this category deserve attention. Joly, Tordeux and co-workers in [12, 21, 22, 23, 24] applied matched asymptotics and provided a full expansion and approximate models for the propagation of waves in thin slots. This work also brought deep insight on the matching principle for a whole class of problems. Dauge, Vial, Costabel and Caloz in [5, 33] studied a thin coating problem in the case of a domain whose boundary contains an angle. An important conclusion of this work is that the ansatz is directly related to the opening of the angle: this case is much more involved than the case of a smooth boundary. Besides in this situation, whether it is possible to derive high order GIBC remains an open question. The works of Tordeux and Vial have inspired further advances in many other situations including patch antennas, see [25], or diffraction by thin wires, see [9, 10, 11].
Open questionsConcerning remaining open questions that we present, we would like to formulate two observations. First, asymptotic problems in a context of time domain scattering have received only few attention so far. Second, theoretical numerical analysis for approximate model in an asymptotic context still remains to be developped for most of the problems already studied from a purely analytical point of view.
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